*MEL 110Development of surfaces

*Prism Made up of same number of rectangles as sides of the baseOne side: Height of the prismOther side: Side of the base

Cylinder RectangleOne side: Circumference of the baseOther side: Height of the cylinder

Pyramid Number of triangles in contactThe base may be included if presentDevelopment is a graphical method of obtaining the area of the surfaces of a solid. When a solid is opened out and its complete surface is laid on a plane, the surface of the solid is said to be developed. The figure thus obtained is called a development of the surfaces of the solid or simply development. Development of the solid, when folded or rolled, gives the solid.hpdfdExamplesT. L.

*Methods used to develop surfacesParallel-line development: Used for prisms, cylinders etc. in which parallel lines are drawn along the surface and transferred to the development.

Radial-line development: Used for pyramids, cones etc. in which the true length of the slant edge or generator is used as radius.

Triangulation development: Complex shapes are divided into a number of triangles and transferred into the development (usually used for transition pieces).

Approximate method: Surface is divided into parts and developed. Used for surfaces such as spheres, paraboloids, ellipsoids etc.

Note:- The surface is preferably cut at the location where the edge will be smallest such that welding or other joining procedures will be minimal.

*Parallel line development: This method is employed to develop the surfaces of prisms and cylinders. Two parallel lines (called stretch-out lines) are drawn from the two ends of the solids and the lateral faces are located between these lines.

*LL= Slant edge.S = Edge of baseH= Height S = Edge of baseH= Height D= base diameterDevelopment of lateral surfaces of different solids.(Lateral surface is the surface excluding top & base)Prisms: No.of RectanglesCylinder: A RectangleCone: (Sector of circle)Pyramids: (No.of triangles)Tetrahedron: Four Equilateral TrianglesAll sides equal in lengthCube: Six Squares.Parallel-line developmentRadial-line development

*R= Base circle radius of coneL= Slant height of coneL1 = Slant height of cut part.Base side Top sideL= Slant edge of pyramidL1 = Slant edge of cut part.DEVELOPMENT OF FRUSTUM OF CONEDEVELOPMENT OF FRUSTUM OF SQUARE PYRAMID

*Project, horizontally, the points of intersection of the cutting plane with the edges.Mark distances 3M, 3N2, b4, d1234ABC1DCube cut by section plane

*2Draw the development of the lower portion of the cone surface cut by a plane. Cone base diameter is 40 mm and height is 50 mm. The cutting plane intersects the cone axis at an angle of 45o and 20 mm below the vertexabcdefg4231o431abcdefgoTFl2b2True lengths b2, 2o obtained by auxiliary view methodoDivide the cone in the top view and project the corresponding generator lines in the front viewDevelop the complete surface of the cone by drawing an arc with radius = length of side generator of cone and length of arc = circumference of cone baseDraw the corresponding generator linesObtain true lengths of o1, o2 etc. by auxiliary view, rotation method OR by projecting onto one of the side generators (which are in true length)Mark the distances (true lengths) o1, o2etc. in the development and join them to get the development of the lower portion of the coneaRadius of cone = RTrue length of (o2, o3) = (o2, o3) etc.

*If R = 2r then = 180, i.e., if the slant height of a cone is equal to its diameter of base then its development is a semicircle of radius equal to the slant height.

*Develop the surface of the symmetrical half of an oblique pyramid with a horizontal regular hexagonal base (side 20 mm and vertex 30 mm above one corner of the base)oabcdabco, dcbcbTrue lengthsocbadTFObtain true lengths of the edges ob and oc by rotation or auxiliary view methodEdge oa is seen in true length in the Front Viewab = bc = cd = side of hexagonal base = 20 mmod and dc can be constructed as they are perpendicular to each otherThe lengths of bc, and ob are known and therefore these distances can be marked with the compassAfter drawing triangles odc and ocb, triangle oba can be completedad

*abcDevelop the surface of the cylinder which is cut as shownab,lc,kd,je,if,hgdefgDivide the base of the cylinder in the top and front views into the a certain number of equal parts (12 here)Develop the surface of the cylinder (rectangle with length p x diameter and height = height of cylinder) and divide it into the same number of equal partsThe projector lines from the top view intersect the cut portion of the cylinder at a, b, c..f.Project these points onto the developed surface15o45opx50100TFhihhahijklf5030o

a, eb, fd, hc, ga, i, db, k, ce, j, hf, l, gi, jk, lOblique square prism

*Oblique prismabcdefabfhighig

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